| studyid_adm | agecalc_adm | height_cm_adm | weight_kg_adm | muac_mm_adm |
|---|---|---|---|---|
| 1 | 16.8 | 79.8 | 11.6 | 150 |
| 2 | 46.1 | 93.0 | 13.6 | 151 |
| 3 | 7.9 | 68.2 | 8.2 | 148 |
| 4 | 38.2 | 95.0 | 12.0 | 138 |
| 5 | 16.1 | 83.0 | 12.0 | 165 |
| 6 | 29.6 | 84.0 | 10.0 | 138 |
Clinical Probability and Diagnostic Testing
2025-03-14
| Disease Present | Disease Absent | Marginal Probability | |
|---|---|---|---|
| Test Positive | TP | FP | PPV = TP/(TP+FP) |
| Test Negative | FN | TN | NPV = TN/(FN+TN) |
| Marginal Probability | Sensitivity = TP/(TP+FN) | Specificity = TN/(FP+TN) |
Formula:
\[P(D|T+) = \frac{P(T+|D) \times P(D)}{P(T+|D) \times P(D) + P(T+|D^c) \times P(D^c)}\]
Where:
Interpretation: The probability of having the disease given a positive test result.
Formula:
\[P(D^c|T-) = \frac{P(T-|D^c) \times P(D^c)}{P(T-|D^c) \times P(D^c) + P(T-|D) \times P(D)}\]
Where:
Interpretation: The probability of not having the disease given a negative test result.
Dataset: Pediatric Sepsis Challenge
Description: Synthetic data for predicting inhospital mortality
Variables:
inhospital_mortality: binary outcome (1 if died, 0 if not)agecalc_adm: child’s age in monthsvaccpneumoc_adm: self reported number of dosesNote: We will use this dataset to illustrate probability and statistical concepts in clinical decision-making for pediatrics.
https://github.com/Kamaleswaran-Lab/The-2024-Pediatric-Sepsis-Challenge
| studyid_adm | agecalc_adm | height_cm_adm | weight_kg_adm | muac_mm_adm |
|---|---|---|---|---|
| 1 | 16.8 | 79.8 | 11.6 | 150 |
| 2 | 46.1 | 93.0 | 13.6 | 151 |
| 3 | 7.9 | 68.2 | 8.2 | 148 |
| 4 | 38.2 | 95.0 | 12.0 | 138 |
| 5 | 16.1 | 83.0 | 12.0 | 165 |
| 6 | 29.6 | 84.0 | 10.0 | 138 |
| 0 | 1 | |
|---|---|---|
| not vaccinated | 180 | 8 |
| vaccinated | 2387 | 111 |
inhospital_mortality ~ sex_adm + agecalc_adm + any_pneumococcal_vaccine + hr_bpm_adm + rr_brpm_app_adm + sysbp_mmhg_adm + diasbp_mmhg_adm + temp_c_adm + muac_mm_adm + weight_kg_adm + momage_adm + glucose_mmolpl_adm
Call:
glm(formula = inhospital_mortality ~ sex_adm + agecalc_adm +
any_pneumococcal_vaccine + hr_bpm_adm + rr_brpm_app_adm +
sysbp_mmhg_adm + diasbp_mmhg_adm + temp_c_adm + muac_mm_adm +
weight_kg_adm + momage_adm + glucose_mmolpl_adm, family = binomial,
data = sepsis_data)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -4.873740 3.867110 -1.260 0.20756
sex_admMale -0.026608 0.194740 -0.137 0.89132
agecalc_adm 0.032629 0.012287 2.656 0.00792 **
any_pneumococcal_vaccinevaccinated 0.123536 0.386640 0.320 0.74934
hr_bpm_adm 0.002770 0.004305 0.643 0.51994
rr_brpm_app_adm 0.006551 0.006639 0.987 0.32374
sysbp_mmhg_adm -0.006290 0.009382 -0.670 0.50260
diasbp_mmhg_adm -0.007546 0.010303 -0.732 0.46390
temp_c_adm 0.120470 0.101572 1.186 0.23560
muac_mm_adm -0.010002 0.008898 -1.124 0.26095
weight_kg_adm -0.155170 0.077929 -1.991 0.04646 *
momage_adm -0.033826 0.017097 -1.978 0.04788 *
glucose_mmolpl_adm 0.073302 0.034800 2.106 0.03517 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 939.02 on 2634 degrees of freedom
Residual deviance: 907.60 on 2622 degrees of freedom
AIC: 933.6
Number of Fisher Scoring iterations: 6
Best threshold: 0.06057769
Sensitivity at best threshold: 0.4122807
Specificity at best threshold: 0.8298294
Pre-test probability (positive): 0.04
Post-test probability (positive): 0.1
Post-test probability (negative): 0.03
A 12-year-old boy presents with 24 hours of abdominal pain. The pain began around the umbilicus and migrated to the right lower quadrant (RLQ) over the last day. He has had two episodes of vomiting and reports poor appetite. His maximum recorded temperature is 37.5°C. On physical examination, he exhibits RLQ tenderness with guarding and positive findings on percussion and hopping tests indicating peritoneal irritation. Laboratory studies reveal a WBC count of 14,500/µL with 85% neutrophils (approximately 12,300/µL). What should we do?
| Criterion | Points | Patient Finding |
|---|---|---|
| Nausea or vomiting | 1 | Yes |
| Anorexia (loss of appetite) | 1 | Yes |
| Pain migration to RLQ | 1 | Yes |
| Fever (≥38°C) | 1 | No (37.5°C) |
| Cough/percussion/hopping tenderness | 2 | Yes |
| RLQ tenderness | 2 | Yes |
| Leukocytosis (WBC > 10,000/µL) | 1 | Yes (14,500/µL) |
| Neutrophilia (ANC > 7,500/µL) | 1 | Yes (12,300/µL) |
Total PAS Calculation:
\[ \text{PAS} = 1 + 1 + 1 + 0 + 2 + 2 + 1 + 1 = 9 \]
A PAS of 7–10 is considered high risk for appendicitis.
| Ultrasound Result | If Appendicitis | If No Appendicitis |
|---|---|---|
| Positive | 85% (True Positive) | 7% (False Positive) |
| Negative | 15% (False Negative) | 93% (True Negative) |
Pretest Probability: 50%
Pre-test Odds:
\[
\text{Odds} = \frac{0.5}{0.5} = 1.0
\]
Likelihood Ratios:
Positive Ultrasound:
\[
\text{Odds}_{post} = 1.0 \times 12.1 = 12.1 \quad \Rightarrow \quad \text{Probability} \approx \frac{12.1}{13.1} \approx 92\%
\]
Negative Ultrasound:
\[
\text{Odds}_{post} = 1.0 \times 0.16 = 0.16 \quad \Rightarrow \quad \text{Probability} \approx \frac{0.16}{1.16} \approx 14\%
\]
An 18-month-old boy, previously healthy and fully immunized, is brought to the clinic in January with a 3-day history of cough, labored breathing, and mild hypoxia (oxygen saturation 87% on room air). He attends daycare where several children have had similar symptoms. On exam, diffuse wheezing and crackles are noted, and his work of breathing is increased. There are no underlying chronic conditions. What should we do?
In winter, epidemiologic data indicate the following approximate prevalence in young children with respiratory illness:
Note: These estimates are based on published surveillance data and may vary by season and region.
The BioFire FilmArray Respiratory Panel is a multiplex PCR assay that detects a wide range of respiratory pathogens with high accuracy. Published performance characteristics from the manufacturer include (the real world accuracy is much lower):
| Pathogen | Sensitivity | Specificity |
|---|---|---|
| Influenza A/B | 90–100% | 99–100% |
| RSV | 95–100% | 90–100% |
| Rhinovirus/Enterovirus | ~93% | ~95% |
| Adenovirus | ~89% | ~98% |
| Human Metapneumovirus | ~94–95% | ~99% |
| Parainfluenza (1–4) | 87–100% | ~99% |
| Mycoplasma pneumoniae | ~84–100% | ~99% |
| Bordetella pertussis | 95–100% | ~99% |
For our case, the panel returns negative for influenza and positive for Mycoplasma pneumoniae.
We will convert our pretest probabilities to post-test probabilities using likelihood ratios.
Given our findings:
A 5-year-old boy, previously healthy and fully immunized, is brought to the clinic with a 2-day history of sore throat and fever (up to 39°C). He attends daycare. On examination, he has:
What should we do?
In school-aged children, epidemiologic data indicate that roughly 15–30% of sore throat cases are due to GAS.
Given his age and daycare exposure, a reasonable baseline pretest probability is about 25%.
The Modified Centor Score allocates points based on clinical findings:
Our Patient’s Score:
Total Score = 5
A score of 5 typically increases the likelihood of GAS to roughly 50–60%. For our purposes, we’ll assume a post-Centor probability of 55%.
The RADT is a quick assay for GAS with high specificity:
From these numbers:
Positive Likelihood Ratio (LR+):
\[ LR_{+} = \frac{0.85}{1-0.95} = \frac{0.85}{0.05} = 17 \]
Negative Likelihood Ratio (LR–):
\[ LR_{-} = \frac{1-0.85}{0.95} = \frac{0.15}{0.95} \approx 0.16 \]
A positive RADT result is highly confirmatory for GAS.
Pretest Probability: 25%
(Pretest Odds = 0.25/0.75 ≈ 0.33)
After Modified Centor Score (assume LR ~4 for a high score):
\[ \text{Post-Centor Odds} = 0.33 \times 4 \approx 1.32 \]
\[ \text{Post-Centor Probability} = \frac{1.32}{1 + 1.32} \approx 57\% \]
We round to approximately 55% as our updated probability.
Using the 55% post-Centor probability as the new pretest probability:
Thus, a positive RADT raises the probability of GAS pharyngitis to approximately 95%.
Sequential Testing Approach:
Management: